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Exploratory Factor Analysis
（探索的因子分析）
Yasuyo Sawaki
Waseda University
JLTA2011 Workshop
Momoyama Gakuin University
October 28, 2011
1
Today’s schedule
Part 1: EFA basics
• Introduction to factor analysis
• Logic behind EFA
• Key steps of EFA
• EFA vs. CFA
(BREAK)
Part 2: EFA Exercise
2
What is factor analysis?
• A multivariate statistical technique developed in the early 1900s
• A method to examine relationships between observed variables （観測変数）and latent factors（潜在因子）
3
Types of factor analysis
• Exploratory factor analysis (EFA; 探索的因子分析)
– Data-driven approach
– Often used in early stages of an investigation
• Confirmatory factor analysis (CFA; 検証的/確認的因子分析)
– Theory-driven approach
– Often used in later stages of an investigation to confirm specific hypotheses
• Combining EFA and CFA
4
Previous applications of EFA in applied linguistics (1)
• Development and validation of survey and assessment instruments
Example 1: Bachman, Davidson, Ryan, & Choi (1995)
• Examining comparability of the factor structures of two English language tests (Cambridge FCE and TOEFL)
• EFAs for each test separately, followed by another EFA run for a combined analysis of the two tests
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Previous applications of EFA in applied linguistics (2)
• Development and validation of survey and assessment instruments
Example 2: Vandergrift, Goh, Mareschal, & Tafagodtari (2006)
• Developing and validating a new survey on L2 learners’ metacognitive awareness and strategy use in listening comprehension
• An initial EFA to finalize survey content, followed by a CFA on a different sample
6
Previous applications of EFA in applied linguistics (3)
• Descriptive use of EFA
Example: Biber, Conrad, Reppen, Byrd, & Helt (2002)
• An EFA of an academic English language corpus to identify linguistic characteristics of various spoken and written registers
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Logic behind factor analysis
• Both EFA and CFA based on the common factor model（共通因
子モデル）
Underlying logic: Variables correlate because they tap into the same construct(s) to certain degrees
Goal: To identify an optimal number of latent factors (factor solution) that describes the pattern of relationships among a set of variables sufficiently well (reproduction of the observed correlation matrix )
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The common factor model
• Decomposing variance of observed variables into two parts:
– Common variance: part of variance influenced by a latent factor shared across different variables (common factors; 共通因子)
– Unique variance: part of variance not explained by the common factors
• Variance explained by factors other than the common factors (unique factors; 独自因子）
• Variance due to measurement error
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EFA: Issues
• Readily available in statistical packages
• Easy to implement, but careful consideration of various issues in different steps of the analysis is required to obtain interpretable and meaningful analysis results
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Key steps of EFA
Step 1: Checking appropriateness of study design and data type for conducting EFA
Step 2: Deciding on the number of factors to extract
Step 3: Extracting and rotating factors
Step 4: Interpreting key EFA results
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A sample scenario
• An EFL speaking and listening test
• Sample size: N=200
• Variables 1-6 for speaking, and Variables 7-12 for listening (4-6 score points available for each variable)
• Expected findings
– There are two latent factors: One each for speaking and listening
– The two factors are correlated with each other because they are both different aspects of L2 language ability
• Software: SPSS (PASW Statistics Version 18)
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Step 1: Checking appropriateness of study design and data type for EFA (1)
Data type: determines the type of correlation matrix to be analyzed
Common examples
• Interval or quasi-interval scales Pearson correlation matrix
• Ordered categorical data
– Dichotomous data tetrachoric correlation matrix
– Polytomous data polychoric correlation matrix
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Step 1: Checking appropriateness of study design and data type for EFA (2)
Number of variables: At least 3 variables needed per factor
Factor
Variable 1
Variable 2
Variable 3
Just identified model
Example 1: 3 variables to identify one factor Number of data points available in a correlation matrix with k=3: k(k+1)/2 = 3(3+1)/2 = 6
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Step 1: Checking appropriateness of study design and data type for EFA (3)
Number of variables: At least 3 variables needed per factor
Over identified model
Factor
Variable 1
Variable 2
Variable 3
Variable 4
Example 2: 4 variables to identify one factor Number of data points available in a correlation matrix with k=4 : k(k+1)/2 = 4(5+1)/2 = 10
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Step 1: Checking appropriateness of study design and data type for EFA (4)
Number of variables: At least 3 variables needed per factor
Under identified model
Factor
Variable 1
Variable 2
Example 2: 4 variables to identify one factor Number of data points available in a correlation matrix with k=2 : k(k+1)/2 = 2(3+1)/2 = 3
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Step 1: Checking appropriateness of study design and data type for EFA (5)
Sample size: Suggestions about sample size in the literature vary
A required sample size depends on many factors such as:
• Strength of correlations between factors and their indicator variables
• Reliability
• Score distribution of variables (e.g., normality)
• Number of factors to extract
SEE: Fabriger et al. (1999), Floyd & Widaman (1995), Tabachnick & Fidell (2007)
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Step 2: Deciding on the number of factors to extract(1)
Various approaches to determining the number of factors
• Approaches based on eigenvalues for the correlation matrix
Eigenvalue（固有値）: Shows how much of the variance of a set of variables can be explained by a factor
• Goodness of model fit（モデルの適合度） : goodness-of-fit statistics available for certain estimation methods (e.g., ML)
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Step 2: Deciding on the number of factors to extract(2)
• Approaches based on eigenvalues for the correlation matrix
Goal: To identify the number of factors with large enough eigenvalues to explain relationships among observed variables
• Kaiser’s criterion (Kaiser, 1960)
• Scree test (Cattell, 1966)
• Parallel analysis (PA; Horn, 1965)
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Step 2: Deciding on the number of factors to extract(3)
• Kaiser’s criterion (Kaiser, 1960)
– The number of factors to extract = the number of factors with eigenvalues exceeding 1.0
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Step 2: Deciding on the number of factors to extract(4)
• Scree plot (Cattell, 1966)
– The number of factors to extract = the “elbow” in the graph
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Step 2: Deciding on the number of factors to extract(5)
• Parallel analysis (PA; Horn, 1965)
– Comparison of eigenvalues obtained from real data against those obtained from multiple samples of random numbers (see Hayton,et al., 2004; Liu& Rijmen, 2008 for sample SPSS and SAS programs)
Steps:
1. Obtain a scree plot for the actual data being analyzed
2. Run a program for PA, which calculates eigenvalues for multiple samples of random numbers; then take the mean of eigenvalues for each component across the PA runs
3. Plot the mean eigenvalues for individual components from the PA over the scree plot for comparing the eigenvalues from the actual data and those from the PA runs
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Step 2: Deciding on the number of factors to extract(6)
• Goodness-of-fit statistics
– Available for certain estimation methods of model parameter estimation
Example: maximum likelihood (ML; 最尤法)
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Step 2: Deciding on the number of factors to extract(7)
Example: maximum likelihood (ML; 最尤法)
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Step 2: Deciding on the number of factors to extract(8)
• Issues of consideration
– Some widely used methods are easy to implement (e.g., Kaiser’s criterion, scree test)
– However, different methods have different disadvantages
• Kaiser’s criterion: Often found to produce misleading results
• Scree plots: Can be difficult to decide where the “elbow” is
• ML estimation: Data have to satisfy distributional assumptions (deviation from normality misleading analysis results)
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Step 2: Deciding on the number of factors to extract(9)
• Issues of consideration (continued)
– What matrix should be analyzed when determining the number of factors? (Brown, 2006; Fabriger et al., 1999)
• Kaiser’s criterion: observed correlation matrix required
• Scree test: possible both on observed and reduced correlation matrices
• What should we do in practice?
– Use multiple methods
– In later steps of the analysis, carefully examine substantive interpretability and parsimony of a given factor solution
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Step 3: Extracting and rotating factors(1)
Choice of a factor extraction method depends on multiple factors such as data type, distribution, and information you need
Common methods used with continuous variables:
Method Distributional assumption
Goodness-of-fit statistics
Principal factor analysis
No assumption imposed
Not available
Maximum likelihood (ML)
Normality assumed Available
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Step 3: Extracting and rotating factors(2)
• Principal component analysis (PCA; 主成分分析)
– NOT based on the common factor model; NOT consistent with the purpose of examining underlying factor structure
– A mathematical data reduction method; NOT based on the common factor model
– More appropriate when, for example, the goal is to create a composite variable out of a larger number of factors
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Step 3: Extracting and rotating factors(3)
Factor rotation
• One factor solution: Results from the initial factor solution can be interpreted
• A solution with multiple factors: Results from the initial factor solution is difficult to interpret Factor rotation (a mathematical transformation) is often conducted
– Orthogonal rotation（直行回転）: Correlations among factors NOT allowed (e.g., Varimax rotation)
– Oblique rotation（斜交回転）: Correlations among factors allowed (e.g., Oblimin rotation; Promax rotation)
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Step 4: Extracting and rotating factors(4)
Factor loadings without
rotation (based on PFA)
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Step 4: Extracting and rotating factors(5)
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Step 4: Extracting and rotating factors(6)
Factor loadings and inter-factor
correlation after Promax rotation
(based on PFA)
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Step 4: Extracting and rotating factors(7)
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Step 3: Extracting and rotating factors(8)
Issues of consideration
• With language data, which type of factor rotation (orthogonal vs. oblique) is generally preferred?
• Should factors be correlated to use an oblique rotation?
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Step 4: Interpreting key analysis results(1)
Key analysis results to focus on:
– Communality （共通性）: The estimated proportion of the variance of a given variable explained by the factors included in the model
– Factor loading（因子負荷量）: A standardized estimate of the strength of the relationship between an observed variable and a latent factor (similar to a standardized regression coefficient)
– Factor correlation（因子間相関）: A “true” correlation between a pair of factors; adjusted for measurement error
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Step 4: Interpreting key analysis results(2)
Communality （共通性）
Communality should be less
than 1.0 for every single
variable.
If communality > 1.0 for any,
the factor solution cannot be
interpreted in a meaningful
way.
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Step 4: Interpreting key analysis results(3)
Calculating communality
Example: When the inter-factor correlation (ϕ21) = .66, the communality of Speaking 3 can be calculated as:
Communality = λ312
+ λ322 + 2λ31 ϕ21λ32
= (.70)2 + (.08)2 + 2(.70)(.66)(.08)
= .490 + .0064 + .074 = .570
(For details about calculating communality, see Brown, 2006, pp. 90-91)
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Step 4: Interpreting key analysis results(4)
Factor loadings and inter-factor
correlation after Promax rotation
(based on PFA)
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EFA vs. CFA (1)
Differences in approaches
• Exploratory factor analysis (EFA; 探索的因子分析)
– Data-driven approach
– Often used in early stages of an investigation
• Confirmatory factor analysis (CFA; 検証的/確認的因子分析)
– Theory-driven approach
– Often used in later stages of an investigation to confirm specific hypotheses
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EFA vs. CFA (2)
Graphic representation of the differences (per Brown, 2006)
Factor 1
Variable 1
Variable 2
Variable 3
Variable 4
Variable 5
Variable 6
Factor 2
EFA
(oblique rotation)
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EFA vs. CFA (3)
Graphic representation of the differences (per Brown, 2006)
Factor 1
Variable 1
Variable 2
Variable 3
Variable 4
Variable 5
Variable 6
Factor 2
CFA
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EFA vs. CFA (4)
Issues of consideration about using EFA and CFA (Jöreskog, 2007)
• Stages of research: exploratory vs. confirmatory phases
• Nature of investigation: “Factor analysis need not be strictly exploratory or strictly confirmatory. Most studies are to some extent both exploratory and confirmatory because they involve some variables of known and other variables of unknown composition.” (Jöreskog, 2007, p. 58)
• Cross-validation of results from exploratory studies by conducting confirmatory analyses on different data sets
(e.g., using randomly-split samples)
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Useful guidelines for conducting EFA
• Factor analysis is often criticized not because of the fundamental weakness of the methodology but because of its misuses in previous factor analysis applications
• Further reading:
– Brown (2006)
– Fabriger et al. (1999)
– Floyd & Widaman (1995)
– Preacher & MacCallum (2003)
– Tabachnick & Fidell (2007)
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References
Bachman, L. F., Davidson, F., Ryan, K., & Choi, I.-C. (1995). An investigation into the comparability of two tests of English as a foreign language. Cambridge, U.K.: Cambridge University Press.
Biber, D., Conrad, S., Reppen, R., Byrd, P., & Helt, M. (2002). Speaking and writing in the university: A multidimensional comparison. TESOL Quarterly, 26, 9-48.
Brown, T. A. (2006). Confirmatory factor analysis for applied research. New York: Guilford.
Cattell, R. B. (1966). The scree test for the number of common factors. Multivariate Behavioral Research, 1, 245-276.
Comrey, A. L., & Lee, H. B. (1992). A first course in factor analysis (2nd ed.). Hillsdale, NJ: Erlbaum.
Fabriger, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. (1999). Evaluating the use of exploratory factor analysis in psychological research. Psychological Methods, 4(3), 272-299.
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References (continued)
Floyd, F. J., & Widaman, K. F. (1995). Factor analysis in the development and refinement of clinical assessment instruments. Psychological Assessment, 7(3), 286-299.
Hayton, J. C., Allen, D. G., & Scarpello, V. (2004). Factor retention decisions in exploratory factor analysis: A tutorial on parallel analysis. Organizational Research Methods, 7(2), 191-205.
Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika, 30, 179-185.
Jöreskog, K. G. (2007). Factor analysis and its extensions. In R. Cudeck & R. C. MacCallum (Eds.), Factor analysis at 100: Historical developments and future directions. Mahwah, NJ: Erlbaum.
Kaiser, H. F. (1960). The application of electronic computers to factor analysis. Educational and Psychological Measurement, 20, 141-151.
Liu, O. L., & Rijmen, F. (2008). A modified procedure for parallel analysis of ordered categorical data. Behavioral Research Methods, 40, 556-562.
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References (continued)
Preacher, K. J., & MacCallum, R. C. (2003). Repairing Tom Swift’s electric factor analysis machine. Understanding Statistics, 2(1), 13-43.
Sawaki, Y. (in press). Factor analysis. In Carol A. Chapelle (Ed.), Encyclopedia of applied linguistics. New York: Wiley.
Tabachnick, B. G., & Fidell, L. S. (2007). Using multivariate statistics (5th ed.). Boston, MA: Pearson.
Vandergrift, L., Goh, C. C. M., Mareschal, C. J., & Tafaghodtari, M. H. (2006). The metacognitive awareness listening questionnaire: Development and validation. Language Learning, 56(3), 431-462.
Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of components to retain. Psychological Bulletin, 99(3), 432-442.
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